Cavity ring down spectroscopy using measured backward mode data

ABSTRACT

In cavity ring-down spectroscopy (CRDS), scattering into the backward mode of a traveling wave ring-down cavity can degrade conventional CRDS performance. We have found that this performance degradation can be alleviated by measuring the backward mode signal emitted from the ring-down cavity, and using this signal to improve the processing for extracting ring-down times from the measured data. For example, fitting an exponential to the sum of the intensities of the forward and backward signals often provides substantially better results for the ring-down time than fitting an exponential to the forward signal alone. Other possibilities include extracting cavity eigenmode signals from the forward and backward signals and performing separate exponential fits to the eigenmode signals. An optical circulator can be used to facilitate measurement of the backward mode signal.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation in part of U.S. patent applicationSer. No. 13/065,270, filed on Mar. 18, 2011, and hereby incorporated byreference in its entirety.

FIELD OF THE INVENTION

This invention relates to cavity ring-down spectroscopy.

BACKGROUND

Cavity ring-down spectroscopy (CRDS) is an analytical technique whereoptical radiation emitted from a passive optical cavity is measured as afunction of time. The decay rate of this emitted radiation is related tothe loss in the cavity (lower loss leads to slower decay). Typically, anexponential decay is fitted to the measured radiation intensity todetermine the ring-down time. Absorption caused by an analyte in thecavity affects the ring-down time, so measuring the ring-down timeamounts to a highly sensitive form of absorption spectroscopy. Theresulting CRDS instruments are widely applicable to various analysisapplications, especially in cases requiring ultra-high sensitivity(e.g., part per billion level).

Ideally, only a single cavity mode is relevant during a ring-downmeasurement, with all other modes having negligible amplitude. Thereason for this is that intracavity loss will tend to have a differenteffect on the decay rates of each of the cavity modes, so a cleansingle-exponential decay can only be obtained for single-modeexcitation.

The cavity in a CRDS instrument can be either a standing wave cavity ora traveling wave cavity. A typical example of a standing wave cavity isa two-mirror cavity where a round trip of the cavity mode entailspropagating on the path between the two mirrors once in each direction.A typical example of a traveling wave cavity is a three mirror ringcavity, where a round trip of the cavity mode entails propagating on apath around the ring cavity (e.g., in a clockwise or counter clockwisedirection).

In CRDS using a traveling wave cavity, one typically distinguishesbetween the forward mode, which is driven by the optical source of theCRDS instrument, and the backward mode which is at the same frequency asthe forward mode, but propagates in the opposite direction. For example,if the forward mode propagates clockwise around a ring cavity, thecorresponding backward mode propagates counter clockwise, and viceversa.

Ideally, the amplitude of the backward mode would be zero, so this modeis usually neglected in conventional accounts of CRDS operation.However, there is some consideration of the backward mode in the art. InU.S. Pat. No. 7,646,485, two ideas are considered. The first idearelates to performing more complicated curve fitting than a simpleexponential to mitigate the effect of backward mode artifacts on CRDSresults. The second idea relates to measuring excitation of the backwardmode by a source aligned to nominally excite only the forward mode toprovide an indication of the quality of the cavity alignment. Adjustmentof the cavity alignment during assembly to minimize the measuredexcitation of the backward mode can be helpful for improving the cavityalignment of the finished instrument.

SUMMARY

In this work, some problems that can arise in connection with excitationof the backward mode of a traveling wave cavity during ring-downmeasurements are identified. Several approaches are also presented foralleviating these problems. To better appreciate the present approach,it is helpful to consider CRDS operation in greater detail.

In cavity ring-down spectroscopy, the optical absorption of a samplewithin an optical cavity is obtained by fitting an exponential to thedecaying intensity of light emanating from the cavity after the(monochromatic) excitation has been turned off. The exponential decay ischaracteristic of the damping of a single mode. With a traveling wavecavity, a pair of degenerate counter-propagating modes is present foreach resonant frequency. Ideally, only one of these modes is excited bythe source but imperfections (such as scattering) can produce a weakcoupling between the primary (ring-down) mode and the backscatteredmode.

During a ring-down, the mode coupling causes periodic exchange of energybetween the modes, in addition to the decay. If the intensity of theprimary mode only is monitored, the waveform is no longer preciselyexponential, leading to a bias in the ring-down time estimate whichdepends on the degree of excitation and relative phase of thebackscattered mode. Since the excitation varies from shot to shot,depending on the precise moment at which the source is turned off, oneeffect of neglecting the backscattered mode is an increase in the noiseor variability of successive ring-down time measurements.

A second effect is found when a spectrum of the cavity loss is measured.Depending on the positions of the scatterers which couple the forwardand backward modes, interference effects can cause the effectivecoupling strength to depend on the frequency of the excitation. The biasand noise introduced by fitting the forward mode alone can thus befrequency dependent, thereby confusing the interpretation of spectra. Inthe case of a 3-mirror ring resonator, if scatterers are on differentmirrors, then the frequency dependence appears as a modulation of theresonator baseline optical loss and noise with a period corresponding tothe reciprocal of twice the spatial separation (the distance between themirrors) modulo the resonator free spectral range.

By solving the equations for the coupled modes during a ring-down, it isfound that under the assumption of fixed point-like weakly scatteringcenters, the sum of the intensities of both modes (i.e., forward andbackward) does decay approximately exponentially, although theintensities of the individual forward and backward modes arenon-exponential. For more general scattering, for example bynearly-resonant atoms, this is no longer precisely true, but it remainsa good approximation if the absorption is weak. Thus, by performing anexponential fit to the sum of intensities of both modes during aring-down, the above-identified deleterious effects can be reduced.

More specifically, a CRDS instrument having a traveling wave cavity isconsidered. The traveling wave cavity has forward and backward modesthat propagate in opposite directions in the cavity. The instrumentincludes an optical source capable of providing optical radiation to thecavity. It also includes an optical detection unit that receives forwardmode and backward mode optical signals from the cavity and provides oneor more electrical detector signals. A processor (e.g., a dataacquisition system) receives the electrical detector signals andprovides a cavity loss derived from the electrical signals as an output.

As seen in greater detail below, there are at least three configurationsfor the optical detection unit. The first configuration has two separatedetectors, one for the forward mode and the other for the backward mode.The second configuration has a single detector at which both the forwardand backward optical signals are detected to provide a sum intensitysignal. In the third configuration, an optical interferometer is addedto the detector unit to transform the forward and backward mode opticalsignals to cavity eigenmode signals (e.g., sine and cosine signals). Thethird configuration can have one or two detectors, depending on whetheror not both interferometer output ports are detected. The path lengthdifference of the interferometer is preferably about a wavelength orless, where the wavelength is set by the optical source and can be anywavelength at which CRDS is possible.

Optionally, an anti-reflection unit can be added to prevent reflectionby the detector(s) from reaching the cavity. Such anti-reflection can beprovided by Faraday isolator(s), or by a less costly combination ofpolarizer and quarter-wave plate if the detector reflection and/orscattering preserves polarization.

As indicated above, some embodiments relate to forming a sum intensitysignal of the forward and backward signals, and to performing anexponential fit to the sum intensity signal to determine the ring-downtime (i.e., the cavity loss). If separate detectors are used for theforward and backward optical signals, the summing is performedelectrically. Analog and/or digital electronics can be used for thisoperation. One option is to provide the detector photocurrents to thesumming junction of a trans-impedance amplifier.

Other embodiments relate to deriving cavity eigenmode signals from theforward and backward mode optical signals. If this is done, moreextensive data processing can be performed. For example, separateexponential fits can be performed for each of the cavity eigenmodesignals to improve the determination of the ring-down time.

Further embodiments make use of an optical circulator to separate sourceradiation from the backward mode signal.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a first embodiment of the invention.

FIG. 2 a shows measured ring-down curves for the forward and backwardmodes in an experiment (first embodiment).

FIG. 2 b shows a scatter plot of measured cavity losses with and withoutcorrections according to the present principles (first embodiment).

FIG. 3 shows a scatter plot of measured cavity losses with and withoutcorrections according to the present principles for a wavelength scan(first embodiment).

FIG. 4 shows a second embodiment of the invention.

FIG. 5 shows a scatter plot of measured cavity losses with and withoutcorrections according to the present principles for a wavelength scan(second embodiment).

FIG. 6 shows an uncorrected measured ring-down curve and residual(second embodiment).

FIG. 7 shows a corrected measured ring-down curve and residual (secondembodiment).

FIG. 8 shows the effect of implementing the correction partway throughcollecting data for a cavity loss scatter plot (second embodiment).

FIG. 9 shows a third embodiment of the invention.

FIG. 10 shows a fourth embodiment of the invention.

DETAILED DESCRIPTION

A first embodiment of the invention involves using two separatephotodiodes to monitor the intensity of light in the forward andbackward modes. For a ring cavity, these may both be convenientlyaccessed from the output mirror as illustrated schematically in FIG. 1.Here a ring cavity is formed by mirrors 102, 104, and 106. A mechanicaltransducer 108 (e.g., a PZT transducer) is connected to mirror 106 suchthat its position can be altered under electrical control, therebyproviding the capability to adjust cavity mode frequencies as needed forCRDS operation. A source 110 provides radiation to the cavity, and itcan be seen from the figure that the forward mode propagates counterclockwise in the cavity of this example. Ring-down detector 112 receivesa signal 150 from the cavity, and it is apparent from the geometry ofFIG. 1 that signal 150 is a forward mode signal. Similarly, it isapparent that backscatter detector 116 receives a backward mode signal160 from the cavity. Light emitted from the cavity follows path 160 onlyif it is propagating clockwise in the cavity (i.e., only if it is thebackward mode). Detectors 112 and 116 are connected to a dataacquisition system 114.

The photocurrents may be individually amplified and the outputs summedusing an analog network or digitally after analog-to-digital conversion.In an alternative realization, the photocurrents may be connected to thesumming junction of a single trans-impedance amplifier. It is importantthat the two channels be matched as well as possible, and for theoutgoing beams to be well-captured by the detectors in order to realizethe full benefits of the method.

Optionally, optical isolation can be provided by isolators 112 a and/or116 a to prevent reflections by detectors 112 and 116 from propagatingto the optical cavity. An optical isolator placed on either or bothbeams 150 and 160 incident on the detectors will also serve to preventaccidental coupling of the forward and backward beams to each otheroutside the cavity. If the detector reflection and scattering preservespolarization, then a cheaper alternative to an isolator with Faradayrotation is a combination of linear polarizer and quarter-wave plate.The polarizer is aligned to the input beam polarization, and then thequarter-wave plate transforms the beam to circular polarization,incident on the detector. Any reflection passes back through thequarter-wave plate, which transforms to linear polarizationperpendicular to the polarizer, which then blocks it.

In FIG. 2 a, forward (top plot) and backward (bottom plot) intensitiesare shown for an average of 1000 ring-down shots collected with light ofa single frequency. The shapes of these waveforms correspond closely tothose predicted by the coupled mode equations for point scatterers. Theresults of performing exponential fitting on the ring-down intensityalone (crosses) and on the sum of both intensities (circles) are shownin FIG. 2 b. In this experiment, the fractional variability in thecavity loss (expressed as a percentage) has been reduced almost by afactor of three, from 0.060% to 0.022%.

In FIG. 3, the excitation frequency is slowly swept from 6251 cm⁻¹ to6252 cm⁻¹ and back to 6251 cm⁻¹ while collecting ring-downs. A smallchange in the cavity loss is expected, due to the frequency dependenceof the mirror coatings, and the change is expected to be approximatelylinear since the sweep is very narrow compared with the coatingbandwidth. The results from fitting the ring-down mode alone (crosses)show an unexpected peak in the loss in the interior of the sweep. Thisis an example of the frequency dependence of the baseline and noisecaused by the interference between (at least) two scatterers. On theother hand, fitting the sum of the mode intensities (circles) shows theexpected smooth change with frequency. The noise on the former spectrumvaries with frequency by a factor of more than three, while that on thelatter is essentially constant.

A second embodiment of this invention involves using a single detectorfor both the forward and backward modes. Two fold mirrors can be addedto a configuration as in FIG. 1 to reflect transmitted backward waves sothat the photon energy of both the forward and the backward waves can becombined and detected by the same photodiode.

FIG. 4 shows an example of this approach. The modification to the systemof FIG. 1 includes two additional fold mirrors 406 and 408. Theseadditional mirrors reflect the backward signal 160 (dotted trace) intothe same ring-down detector 402 that detects the forward signal 150. Thetwo added mirrors are aligned so that the backward and forward beams arenearly parallel to each other. As a consequence only one focus lens 404is needed to collect both beams to the ring-down detector 402.Optionally, isolation can be provided by an isolator 410, in a similarmanner as described in connection with FIG. 1. The system of FIG. 4 isan illustrative example, and practice of the invention does not dependcritically on details of the fold mirrors or the lens, such as theirnumber and arrangement, provided that both forward and backward signalsreach the same detector. It is also not critical whether or not the twobeams interfere at the detector, provided that the detector is linearover the relevant optical intensity range.

In FIG. 5, the laser wavelength is swept between 6249 cm⁻¹ and 6253 cm⁻¹while collecting ring-downs. The overall change in loss is due to thefrequency dependence of the mirror coatings, and the change is expectedto be approximately linear since the sweep is very narrow compared withthe coating bandwidth. The trace marked with crosses shows the frequencydependence of the loss before the modification was implemented. A ripplewith a period of ˜1.2 wavenumber is clearly seen. The noise on thespectrum varies with frequency by a factor of more than three. With thepresent approach, this ripple is almost completely removed by detectingthe combined photon energy from both the forward and backward waves at asingle detector (trace marked with circles). In addition the noise,reduced by a factor of ˜3, is essentially constant throughout the wholescan region.

FIG. 6 shows the detector signal (solid line) of a single ring-down ofonly the forward output beam. The residual of a pure single-exponentialfit multiplied by a factor of 150 is shown with a dashed line.

FIG. 7 shows the detector signal (solid line) of a single ring-down offorward and backward output beams incident on the same detector (e.g.,optical summing). The residual of a pure single-exponential fitmultiplied by a factor of 150 is shown with a dashed line. The residualis significantly lower here than on FIG. 6.

FIG. 8 shows a time-series of ring-down times. Early times are when onlythe forward beam is incident on the detector (as in FIG. 6). Later timesare when both forward and backward output beams are incident on thering-down detector (as in FIG. 7). The difference in shot-to-shotvariation of the ring-down times is evident.

The preceding description has considered the use of a combined forwardand backward mode signal to improve CRDS measurements. It is alsopossible to analyze the forward and backward mode signals moreprecisely. In particular, cavity eigenmode signals can be obtained fromthe forward and backward mode signals. The following descriptionprovides an example of this approach in a relatively simple case wherethe cavity eigenmodes are sine and cosine modes.

In the absence of backscatter coupling, the forward and backward opticalwaves in the ring-down optical resonator are degenerate: they haveidentical resonant optical frequencies and ring-down times (at eachresonant frequency). When backscatter coupling exists within thering-down optical resonator, the degeneracy between forward and backwardoptical waves is broken. In addition, the normal modes of the ring-downresonator are not pure forward and backward traveling waves; they are“sine” and “cosine” standing waves, which are linear combinations of theforward and backward waves. The field amplitudes of these wavestransform as:

${{cosine}\mspace{14mu}{mode}} = {\frac{1}{\sqrt{2}}\left\lbrack {{\left( {{forward}\mspace{14mu}{mode}} \right){\mathbb{e}}^{{+ {\mathbb{i}}}\;\varphi}} + {\left( {{backward}\mspace{14mu}{mode}} \right){\mathbb{e}}^{{- {\mathbb{i}}}\;\varphi}}} \right\rbrack}$${{sine}\mspace{14mu}{mode}} = {\frac{1}{\sqrt{2}}\left\lbrack {{\left( {{forward}\mspace{14mu}{mode}} \right){\mathbb{e}}^{{+ {\mathbb{i}}}\;\varphi}} - {\left( {{backward}\mspace{14mu}{mode}} \right){\mathbb{e}}^{{- {\mathbb{i}}}\;\varphi}}} \right\rbrack}$The relative phase, φ, depends on the physical locations of the pointscatterers within the resonator. These sine and cosine normal modes eachindividually have purely exponential ring-down behavior, and a set offrequency resonances. However, the resonant frequencies and ring-downtimes are not identical between the modes.

For small backscatter coupling, the resonant frequencies forcorresponding mode numbers are slightly shifted from each other and fromthe degenerate frequency in the absence of coupling. If the real part ofthe coupling (the power-loss part) is negligible, then the ring-downtimes are negligibly different, and the power exponential decays areindistinguishable. This is the case mitigated by the two summing methodsdescribed above, using either two separate detectors for forward andbackward optical beams or directing both beams onto one detector. Thesemethods effectively add the powers of the two normal modes, sine andcosine, together, since the sum of forward and backward waves equals thesum of sine and cosine waves equals the total power circulating withinthe resonator, as measured by the power emanating from the cavity outputmirror.

If the real part of the backscatter coupling (the power-loss part) isnot negligible, then the ring-down times of the sine and cosine modeswill differ significantly, and the sum of the signals (the total poweremanating from the output mirror) will be the sum of two exponentialdecays, a bi-exponential. Only by observing either normal modeindividually will the decay be a pure single exponential. In addition,both ring-down times must be known to extract the effect of thebackscatter coupling from other optical losses, such as resonator losses(mirror transmission, or scattering that does not result in backscattercoupling) and analyte absorption and scattering.

An interferometer, with appropriate optical delay, will transform theforward and backward output beams into the sine and cosine beams. Thisinterferometer can be placed next to the output mirror, as shown in FIG.9. In this figure, the two output optical beams 150 and 160 from aring-down cavity 902 are directed along nearly equal paths by mirrors910 and 912, preferably differing by less than one optical wavelength.The path length difference, ΔL, determines the phase, φ=2πΔL/λ. Thebeam-splitter 914 is preferably 50%/50%. At least one beam from thebeam-splitter is directed to a detector 916 (either the sine or cosinebeam can be used). The path length difference ΔL needs to be stable to asmall fraction of a wavelength, e.g. a few nm for visible and near-IR.This can be accomplished by attaching the interferometer mirrors 910 and912 to a stable substrate such as AlN or CuW and controlling theatmosphere, or containing the beams inside a solid prism or prisms. Thisexample includes optional optical wedges 920 and 922 on each beam pathto adjust the path lengths independently. Optionally, isolation can beprovided by adding isolators (not shown) in the paths of beams 150and/or 160 between beam splitter 914 and detectors 916 and 918, in asimilar manner as described in connection with FIG. 1.

The path length difference ΔL can be tuned by adjusting the prism(s) orwedge(s) temperature(s) or stress (e.g. with a PZT) or translating thewedges as is one of the typical methods to tune an interferometer ofthis type. ΔL should be adjusted to maximize the difference in ring-downdecay times observed on the two detectors 916 and 918 (or to an extremumof ring-down time if only one detector is present). By convention, thesine beam has maximum ring-down time and the cosine beam has minimumring-down time. This is because, in the case of a single pointscatterer, the sine wave has a field node (zero field thus minimalinteraction) at the location of the scatterer in the resonator, and thecosine wave has an anti-node there (maximum field thus maximumscattering loss). If the scattering properties of the resonator changeover time, ΔL can be readjusted by a feedback loop that continuously ordiscretely re-maximizes the ring-down time difference (or extremum). Thebandwidth over which ΔL is set properly is approximately λ²/(rΔL) wherer is the permissible deviation of (φ/2π). If ΔL<λ₀ for λ₀ centralwavelength, then the bandwidth essentially extends from λ>>λ₀ to λ≧ΔL.

If the difference in ring-down times between sine and cosine beams issignificant, but only the summed beam is observed (either single or dualdetector), then the decay signal can be fit to a bi-exponential witheither variable or fixed difference between the two decay rates. Thefixed difference can be predetermined by characterizing the ring-downeither using a temporary interferometer to measure the sine and cosinebeams separately, or by carefully fitting the summed signal with abi-exponential. If the difference between decay times is expected tochange slowly, then an averaging loop in the analysis can be used topermit that difference to vary slowly over time, thus partially limitingthe degrees of freedom of the fit and reducing its shot-to-shotvariability in the short term.

FIG. 10 shows an embodiment where an optical circulator is used tofacilitate separation of the backward mode signal from the incidentsource radiation. Here an optical circulator 1002 is present in thesystem, which has the characteristic behavior that light incident onport 1 is emitted from port 2, and light incident on port 2 is emittedfrom port 3. Thus light from laser source 110 is incident on mirror 102of the resonant cavity, and drives the forward propagating mode withinthe cavity (heavy solid arrows). Forward propagating light emitted fromthe cavity is received at ringdown detector 1006 (an optional foldmirror 1004 is shown in this example, but has no significant effect onoperation).

Within the cavity formed by mirrors 102, 104 and 106, scattering orother processes can lead to the propagation of a backward mode (lightdotted arrows). Such backward propagating light will be incident on port2 of the circulator when it couples out of the resonant cavity, and willtherefore be emitted from port 3 of the circular, to be received byringdown detector 1008. Thus circulator 1002 separates the sourceradiation from the backward mode signal. Ringdown detectors 1006 and1008 are connected to control/data acquisition system 1010.

This approach for implementing the back scattered wave correctionadvantageously eliminates the need for free space optical alignment inthe backward beam path. The photocurrents from detectors 1006 and 1008may be individually amplified and the outputs summed using an analognetwork or digitally after analog-to-digital conversion. In analternative realization, these photocurrents may be connected to thesumming junction of a single trans-impedance amplifier. It is preferredthat the two detection channels be matched as well as possible, and forthe outgoing beams to be well-captured by the detectors in order torealize the full benefits of the method.

An optical circulator for separating the source radiation from thebackward mode optical signal can also be used in connection with any ofthe previously described embodiments.

The invention claimed is:
 1. Apparatus for performing cavity ring-downspectroscopy (CRDS), the apparatus comprising: a traveling-wave cavityhaving forward and backward optical modes that propagate in oppositedirections in the traveling-wave cavity; an optical source capable ofproviding optical source radiation to the optical cavity; an opticaldetection unit that receives a forward mode optical signal from thecavity and a backward mode optical signal from the cavity and providesone or more electrical detector signals derived from the forward modeand backward mode optical signals; an optical circulator configured toseparate the source radiation from the backward mode optical signal; anda processor that receives the electrical detector signals and provides acavity loss derived from the one or more electrical signals; wherein theprocessor derives a ring-down time from the electrical detector signalsand wherein the cavity loss is derived from the ring-down time.
 2. Theapparatus of claim 1, wherein the optical detection unit comprises twodetectors that separately receive the forward mode and backward modeoptical signals.
 3. The apparatus of claim 1, wherein the opticaldetection unit comprises a single detector that receives both theforward mode and backward mode optical signals, whereby the singledetector provides a sum intensity signal.
 4. The apparatus of claim 1,wherein the optical detection unit comprises an optical interferometercapable of transforming the forward mode and backward mode opticalsignals at an interferometer input to cavity eigenmode optical signalsat an interferometer output.
 5. The apparatus of claim 1, furthercomprising an anti-reflection unit disposed between the cavity and theoptical detection unit to substantially prevent optical reflections bythe optical detection unit from reaching the cavity.
 6. The apparatus ofclaim 5, wherein the anti-reflection unit comprises an optical isolatoror a combination of a quarter-wave plate and a linear polarizer.
 7. Amethod for performing cavity ring-down spectroscopy (CRDS), the methodcomprising: providing a traveling-wave cavity having forward andbackward optical modes that propagate in opposite directions in thetraveling-wave cavity; providing optical source radiation to the cavity;receiving a forward mode optical signal from the cavity; receiving abackward mode optical signal from the cavity; providing an opticalcirculator configured to separate the source radiation from the backwardmode optical signal; determining a cavity loss from the forward mode andbackward mode optical signals; and providing the cavity loss as anoutput; wherein a ring-down time is derived from the forward mode andbackward mode signals and wherein the cavity loss is derived from thering-down time.
 8. The method of claim 7, further comprising deriving asum intensity signal from the forward mode and backward mode opticalsignals.
 9. The method of claim 8, further comprising performing anexponential fit to the sum intensity signal to determine the cavityloss.
 10. The method of claim 8, wherein output signals from twoseparate detectors are summed to provide the sum intensity signal. 11.The method of claim 10, wherein the output signals from the twodetectors are separately amplified and then summed digitally or withanalog electronics.
 12. The method of claim 10, wherein the outputsignals from the two detectors are amplified together using a summingamplifier.
 13. The method of claim 7, further comprising deriving cavityeigenmode optical signals from the forward mode and backward modeoptical signals with an optical interferometer.
 14. The method of claim13, further comprising performing separate exponential fits to thecavity eigenmode optical signals to determine the cavity loss.
 15. Themethod of claim 13, further comprising setting a path length differenceof the interferometer to be less than or about equal to a wavelength ofan optical source that provides the optical radiation to the cavity.